\(\Lambda_{24}\) Leech lattice-shell code[1] 

Description

Spherical code whose codewords are points on the \(\Lambda_{24}\) Leech lattice normalized to lie on the unit sphere. The minimal shell of the lattice yields the \((24,196560,1)\) code.

Protection

Smallest-shell code yields an optimal solution to the kissing problem in 24D [2]. This code saturates the Levenshtein bound [35][6; pg. 337] and is unique up to equivalence [2].

Parent

Cousins

References

[1]
J. Leech, “Notes on Sphere Packings”, Canadian Journal of Mathematics 19, 251 (1967) DOI
[2]
E. Bannai and N. J. A. Sloane, “Uniqueness of Certain Spherical Codes”, Canadian Journal of Mathematics 33, 437 (1981) DOI
[3]
V. I. Levenshtein, “On bounds for packings in n-dimensional Euclidean space”, Dokl. Akad. Nauk SSSR, 245:6 (1979), 1299–1303
[4]
V. I. Levenshtein. Bounds for packings of metric spaces and some of their applications. Problemy Kibernet, 40 (1983), 43-110.
[5]
A. M. Odlyzko and N. J. A. Sloane, “New bounds on the number of unit spheres that can touch a unit sphere in n dimensions”, Journal of Combinatorial Theory, Series A 26, 210 (1979) DOI
[6]
J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups (Springer New York, 1999) DOI
[7]
Andreev, N. N. Location of points on a sphere with minimal energy. (Russian) Tr. Mat. Inst. Steklova 219 (1997), Teor. Priblizh. Garmon. Anal., 27–31; translation in Proc. Steklov Inst. Math. 1997, no. 4(219), 20–24
[8]
H. Cohn and A. Kumar, “Universally optimal distribution of points on spheres”, Journal of the American Mathematical Society 20, 99 (2006) arXiv:math/0607446 DOI
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Zoo Code ID: leech_shell

Cite as:
\(\Lambda_{24}\) Leech lattice-shell code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/leech_shell
BibTeX:
@incollection{eczoo_leech_shell, title={\(\Lambda_{24}\) Leech lattice-shell code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/leech_shell} }
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Cite as:

\(\Lambda_{24}\) Leech lattice-shell code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/leech_shell

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/classical/spherical/lattice_shell/leech_shell.yml.